The Immersed Interface Method Using a Finite Element Formulation Zhilin

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254 Z. Li /Ap pli ed Numerical Mathematics 27 (1998) 253-267

A

u(x)

x j

oL

x j+l

F ig . 1 . A d i ag ram sh o ws wh y a f i n i te e l em en t m e th o d can n o t b e seco n d o rd er accu ra t e i n t h e i n f in i t y n o rm i f t h e i n t e r f ace

is no t a g r id po in t .

[u]~ de j l im

u ( x ) -

l i m

u ( x ) =

O, (1.3)

x ~ ? x ~ ;

[~Ux]c~

de=f lim /3( x)

u'(x) --

l im /3(x)

u ' (x )

= 0. (1.4)

X ----+C~? X - ~O~

T he so l u t i on o f (1. l ) i s typ i ca l l y non - s m o o t h a t t he i n te r f ace s i f / 3 (x ) ha s a f i n it e j um p a t e ach i n t e r f ace .

T h e p r o b l e m c a n b e s o l v e d b y b o t h f i n it e d if f e r e n c e m e t h o d s a n d f in i te e l e m e n t m e t h o d s . T h e

i m m erse d in t e r f ace m e t h od ( I IM ) [6 , 7 , 9 -11 ] i s an e f f i c i en t f i n it e d i f f e r ence app roac h fo r in t e r f ace

p rob l em s w i t h d i scon t i nu i t ie s and s i ngu l a ri t ie s . T he so l u t ion ob t a i ned f ro m t he I IM i s typ i ca l l y s econ d

o rde r a ccu ra t e in t he i n f i n it y no rm reg a rd l e s s o f t he r e l a ti ve pos i t ion be t w e en t he g r i d po i n t s and

t he i n t e r f ace s . H ow eve r , fo r t w o o r h i ghe r d i m ens i ona l p rob l em s , t he r e su l t i ng l i nea r sy s t em ob t a i ned

f r o m t h e I I M m a y n o t b e s y m m e t r i c p o s it iv e d e f i n it e .

I f t he f in i te e l em en t m e t hod w i t h t he s t anda rd l i nea r ba s i s i s u sed fo r (1. l ) w i t h p re se nce o f i n te r f ace s ,

s econd o rde r a ccu ra t e so l u t i ons can s t i l l b e ob t a i ned i f t he i n t e r f ace s l i e on t he g r i d po i n t s . T h i s c an

be p roved s t r i c t l y i n one d i m ens i ona l space . F o r h i ghe r d i m ens i ona l p rob l em s , t he ana l y s i s i s u sua l l y

g i ven i n an i n t eg ra l no rm w h i ch i s w eak e r than t he i n f in i t y no rm , s ee [1 ,3 , 4, 13 ], e t c . I f any o f t he

i n t e r f ace s i s no t a g r i d po i n t , t hen t he so l u t i on ob t a i ned f rom t he f i n i t e e l em en t m e t hod i s on l y f i r s t

o rde r a ccu ra t e in t he i n f i n i ty no rm , s ee F i g . 1 . F o r t w o o r h i ghe r d i m en s i ona l p rob l em s , i t i s d i ff i cu lt

and cos t l y to c ons t ruc t a body - f i t ti ng g r i d so t ha t t he i n t e r f ace a l i gns w i t h t he t r iangu l a t ion , e spec i a l l y

f o r m o v i n g i n te r f a c e p r o b le m s .

I n t h is p a p e r , w e t r y t o d e v e l o p a n u m e r i c a l m e t h o d w h i c h m a i n t a i n s t h e a d v a n t a g e s o f t h e s i m p l e

g r i d s t ruc t u re o f t he f i n i t e d i f f e r ence m e t hod and t he n i ce t heo re t i c a l p rope r t i e s o f t he f i n i t e e l em en t

m e t hod . T he i dea i s t o t ake a s i m p l e C a r t e s i an g r i d , fo r exam pl e , a un i fo rm g r i d , and m od i fy t he

bas i s func t i ons so t ha t t he i n t e r f ace j um p r e l a t ions a r e s a t i s fi ed . W i t h t he s i m p l e g r i d , o r t r i angu l a t ion ,

t he f in i te e l em en t m e t ho d co r r e sp onds t o a f i n i te d i f f e r ence m e t ho d i n w h i ch t he r e su l t ing l i nea r sy s -

t e m o f e q u a t i o n s i s s y m m e t r i c p o s it iv e d e f in i te . B y c h o o s i n g m o d i f i e d b a si s f u n c t i o n s , s e c o n d o r d e r

a c c u r a c y i s a c h i e v e d in t h e i n f in i ty n o r m f o r o n e d i m e n s i o n a l p r o b l e m s . I n h o m o g e n e o u s j u m p c o n -

d i ti o n s t h e n c a n b e t a k e n c a r e o f e a s i ly b y a d d i n g s o m e c o r r e c t i o n t e r m s a c c o r d i n g to t h e i m m e r s e d

i n t er f a c e m e t h o d . W e a l so p r o p o s e s o m e n u m e r i c a l m e t h o d s f o r c o n s t r u c ti n g b a si s f u n c t i o n s f o r tw o

d i m e n s i o n a l p r o b l e m s i n v o l v in g i n t e r fa c e s . W h i l e s e c o n d o r d e r c o n v e r g e n c e is p r e s e r v e d i n t h e e n -

e r g y n o r m f o r t h o s e m e t h o d s , t h e c o n v e r g e n c e o f t h o s e m e t h o d s i n t h e i n f in i ty n o r m i s s til l u n d e r

inves t iga t ion .



Z. Li /Ap pl ied Numerical Mathematics 27 (1998) 253-267 255

2 . M o d i f i c a t i o n o f t h e l i n e a r b a s i s

D ef i n e t h e s t an d a r d b i l i n ea r f o r m

a ( u , v ) = / ( / 3 ( x ) u ' ( x ) v ' ( x ) + q ( x ) u ( x ) v ( x ) ) d x , u ( x ) , v ( x ) E

H(~(0, 1), (2.5 )

0

w h er e H ~ ( 0 , 1 ) i s t h e S o b o l e v s p ace . T h e s o l u t i o n o f t h e d if f e r en t ia l eq u a t i o n u ( x ) ~ H ~ ( 0 , 1 ) i s a l s o

t h e s o l u t i o n o f t h e f o l l o w i n g v a r i a t io n a l p r o b l em :

a ( u ,v ) = ( f , v ) = / f ( x ) v ( x ) d x , V v E H ~ (O, 1) . (2.6)

.J

0

W i t h o u t l o s s o f g en e r a l i ty , w e a s s u m e t h a t t h e r e i s o n l y o n e i n t e r f ace a i n t h e i n t e r v a l ( 0 , 1 ) . I n t eg r a t i o n

b y p a r t s o v e r t h e s ep a r a t ed i n t e r v a ls ( 0 , a ) an d ( a , 1 ) y i e l d s

OL

= / { - ( / 3 u ' ) ' + q u - f } v + / 3 - u z v (2.7)

0

+ / { - ( / 3 u ' ) ' + q u f } v +~ + "+

- / 3 ~ . v . 2 . 8 )

O~

T h e s u p e r s c r i p t s - an d ÷ i n d i ca t e t h e l i m i t in g v a l u e a s x ap p r o ach es a f r o m t h e le f t an d ri g h t,

r e s p ec t i v e l y , an d u x = u ' . R e ca l l th a t v - = v + f o r an y v i n H d , i t f o l l o w s th a t th e d i f f e r en ti a l

eq u a t i o n h o l d s i n each i n t e r v a l an d t h a t

u + u 0 , + +

- - = = / 3 U x - / 3 - u : ; = 0 ,

w h e r e w e h a v e d r o p p e d t h e s u b s c r i p t a i n th e j u m p s s i n c e th e r e i s o n l y o n e i n t e r fa c e . T h e s e r e la t io n s

a r e t h e s am e a s i n ( 1 . 3 ) , ( 1 . 4 ) , w h i ch i n d i ca t e s t h a t t h e d i s co n t i n u i t y i n t h e co e f f i c i en t / 3 ( x ) d o es n o t

c a u s e a n y t r o u b l e f o r th e t h e o r e t ic a l a n a l y s i s o f t h e F E M a n d t h e w e a k s o l u ti o n w i l l s a ti s fy t h e j u m p

condi t ions (1 .3) , (1 .4) .

N o w l e t u s t u r n o u r a t t en t io n t o t h e n u m er i c s . F o r s i m p l i c it y , w e u s e a u n i f o r m g r i d x i = i h ,

i - - 0 , 1 ~ . . . , N , w i t h x 0 = 0 ,

X N

= 1 and h =

1 / N

i n o u r d i s cu s s i o n . T h e s t an d a r d l i n ea r b a s i s

f u n c t i o n s a t i s f i e s

1 , i f i = k , ( 2 . 9 )

¢ i ( x k ) = 0 , o th e rw i se .

T h e s o l u t i o n

Uh(X)

i s a s p ec i f i c l i n ea r co m b i n a t i o n o f t h e b a s i s f u n c t i o n f r o m t h e f i n i t e d i m en s i o n a l

s p a c e

Vh :

Vh = Vh: Vh = ~ i ¢ i ( x ) , ( 2 . 1 0 )

i=1

a(uh, Vh) = (f , Vh),

f o r V v h E

Vh.

(2 .11)



256 Z.

Li /Ap pl ied Numerical Mathematics 27 (1998) 253-267

I f an i n t e r f ace i s n o t o n e o f g r i d p o i n t s x i , u s u a l l y t h e s o l u t i o n

U h

i s o n l y f i r s t o r d e r a ccu r a t e i n t h e

i n fi n it e n o r m , s ee F i g . 1 . T h e p r o b l em i s t h a t s o m e b as i s f u n c t i o n s w h i ch h av e n o n - ze r o s u p p o r t n ea r

t h e i n t e r f ace d o n o t s a t i s f y t h e n a t u r a l j u m p co n d i t i o n ( 1 . 4 ) a t t h e i n t e r f ace .

T h e s o l u t i o n i s t o m o d i f y t h e b a s i s f u n c t i o n s i n s u ch a w ay t h a t n a t u r a l j u m p co n d i t i o n s a r e s a t i s f i ed :

1 , i f i = k , ( 2 . 1 2 )

¢ i ( x k ) = O, o t h e rw i s e,

[¢i] = O, (2.1 3)

[fl¢'~] = O. (2 .1 4)

O b v i o u s l y , i f

x j < a < x j + j ,

t h e n o n l y C j a n d C j + l n e e d t o b e c h a n g e d t o s a t is f y t h e s e c o n d j u m p

c o n d i ti o n . U s i n g a n u n d e t e r m i n e d c o e f f i c ie n t m e t h o d , w e c a n c o n c l u d e t h a t

O 0 ~< x < x i _ l ,

x - x j - i

h ' xj _l ~ x < X j,

C j ( x ) = x j -

D + 1 , x j <~ x < a ,

( 2 . 1 5 )

o x j + l - x )

a ~ X < Xj+ I,

D

O,

X j+ l ~< X ~< 1,

w h e r e

n = 4 0 , ~ - = 1 , ~ + = 5 , a = 2 /3

n=40 , 13- :5 , 13+=1, x=2 /3

1

0 . 8

0 . 6

0 . 4

0 . 2

0 . 6 0 . 6 5 0 . 7

0 .8

0 . 6

0 .4

0 . 2

0 . 6 0 . 6 5 0 . 7

0 . 8

0 . 6

0 . 4

0 . 2

0

n=40, 13-=1 , ~+=100, o~=2 /3 n=40 , 13-=100 , 13+=1, z=2 /3

0 . 8

0 .6

0 . 4

0 .2

0

0 . 6 0 . 6 5 0 . 7 0 . 6 0 . 6 5 0 . 7

Fig. 2. Plot of some basis function near the interface with dif ferent/3- and/3+.



Z . L i /A pp l i e d Num er i ca l Mathem at i cs 27 (1998) 253-2 67

257

a n d

Z - Z +

_ / 3

P

= ~ -T , D = h /3+ (x j+ l - ~ ) ,

(2 .16)

O, O ~ X < X j ,

X - - X j

D ' x j ~ x < o : ,

(~j+l(X) = p ( x - X j + l ) -~ - 1 c~ <<. x < x j + l ,

D

x j + 2 - - x

h ' X j + l ~ X ~ X j + 2 ,

O,

X j+ 2 <<. X <<. 1.

(2 .17)

F i g . 2 show s seve ra l p l o t s o f t he m od i f i ed ba s i s func t i ons

C j ( X ) ,

0j+ l (X) , a n d s o m e n e i g h b o r i n g

bas i s func t i ons , t ha t a r e the s t anda rd ha t func t i ons . A t t he i n t e r f ace , w e c an see c l ea r l y t he k i nk i n t he

bas i s func t i on w h i ch r e f l e c t t he na t u ra l j um p cond i t i ons .

3 T h e t h e o r e t ic a l a n a l y s i s

In t h is s ec t i on , w e p ro ve t ha t t he so l u t ion ob t a i ned f ro m t he f i n it e e l em en t m e t h od w i t h t he m od i f i ed

bas i s func t i on i s s econd o rde r a ccu ra t e i n t he i n f i n i t e no rm .

F o r t he s ake o f c l ean and conc i se p roo f , w e de r i ve t he t heo re t i c a l ana l y s i s fo r t he s i m p l e m ode l :

- / 3 ( x ) u " = f ( x ) , f ( x ) E C [0 , 1], 0 <~ x ~< 1, (3 .18 )

u(0 ) = O, u( 1) = 0 , (3 .19)

w i t h

/3(x) = { / 3 - , i f O ~< x < c~,

/3+ , i f c ~ < x ~ < l ,

w h e r e / 3 - a n d / 3 + a r e t w o c o n s t a n t s . T h e s o l u t i o n u ( x ) E H i sa t i s f i e s t he na t u ra l j um p cond i t i ons

a t c~. I f t he va l ue o f t he so l u t i on a t c~ i s know n , s ay uc~ , t hen t he p r ob l em i s eq u i va l en t t o t he fo l l ow i ng

t w o s e p a r a t e d p r o b le m s :

- / 3 - u " = f ( x ) , 0 ~ < x < o ~ , an d u ( 1 ) = 0 .

~ (0 ) = 0 , ~ (~ ) = ~ , ~ (~ ) = ~ ,

T h e r e f o r e f r o m t h e r e g u l a r i t y t h e o r y w e k n o w t h a t

u ( x ) E

C2[0 , 1] i n each sub-domain and uz-~ =

l im x ~ a -

u " ( x )

a n d u +x = l i m x ~ +

u " ( x )

are f in i t e . We def ine

IL~ llo~ = - m a x I~xxl, I x~l, sup

lUxxl,

sup I~xxl , (3 .20)

0<x<a c~<x<l

w h i c h i s b o u n d e d .

3.1. The in terpolant o f the solut ion

A s i n t he s t anda rd F E M ana l y s i s , an i n t e rpo l a t i ng func t i on o f t he so l u t ion p l ays an i m po r t an t ro l e in

t he e r ro r ana l y s i s . I n th i s subsec t i on , w e w i l l de f i ne t he p i ecew i se li nea r func t i on i n the space ~ w h i ch



258 Z.

L i / A p p l i e d N u m e r i c a l M a t h em a t i c s 2 7 (1 9 9 8 ) 2 5 3 - 2 6 7

a l s o i n t e r p o l a t e s u ( x ) a t t h e n o d e p o i n t s . A s s u m i n g t h a t xj <~ o~ < xj + l, w e d e f i n e a n i n t e r p o l a n t o f

u ( x )

a s f o l l o w s :

X i + l - - X X - - X i

h u (x i ) + - - -h -u (x i+ l ) , i # j , x~ <~ x <~ X~+l ,

u i ( x ) = u (x j ) + t ~ (x - u ( x j ) ) , x j ~< x < o~, (3 .2 1)

u(X j+ l) + ~ p ( x - x j+~ ), o~ <~ x <~

xj+, ,

w h e r e

9 - ~ x j + l ) -

~(~)

p - f l+ , t~ = . ( 3 .22 )

o ~ - x j - p o ~ - X j + l )

I t i s e a s y t o v e r i f y t h a t

U l ( X i ) = u ( x i ) , i = 0 , 1 , . . . , N - 1 , ( 3 . 2 3 )

[ui] = O, [ /3u}] = O, (3. 24 )

a n d h e n c e

ux(x ) E Vh.

B e f o r e g i v in g a n e r ro r b o u n d f o r ] ] u i ( x ) - u ( x ) ] ] , ¢ , w e n e e d t he f o l lo w i n g

l e m m a w h i c h g i v e s t h e e r ro r e s t i m a t e s f o r t h e f ir st d e r i v a t i v e s o f u i ( x ) a p p r o x i m a t i n g u ' ( x ) .

L e m m a 3 . 1 .

G iven u i ( x ) as de f ined in

(3 .21 ) ,

the fol lo win g inequali t ies hold:

OZ - - X j - - p ( O - - X j + I ) ~ m i n { ½ h , ½ h P } ,

I ,~ - ~ ; I ~ < c I l u l l ~ h ,

I p ~ - ux+l ~< C Ilu ll~ h,

w h e r e

2 max{ 1 , p}

C -

m i n { 1 , p }

( 3 . 2 5 )

( 3 . 2 6 )

( 3 . 2 7 )

( 3 . 2 8 )

S o C a c t s l i ke a c o n d i t io n n u m b e r f o r t h e in t e r f a c e p r o b l e m .

P r o o f . I t i s o b v i o u s th a t

{ 0~ - - X j - - p ( ( y - X j + l ) ~ p ( x j + l - o L ) ~ ½hP, i f ~ - x j < ½h,

OL - - X j - - p (O L - - X y + I ) ~ O - - X j

~ ½ h, if c~ -

x j > ½h,

w h i c h c o n c l u d e s t h e f i rs t i n e q u a l i t y . U s i n g t h e T a y l o r e x p a n s i o n a b o u t c~, w e h a v e

- x j - p c ~ - x y + l )

= ~ + + ~ + x j + ~ - ~ ) + ½ ~ x x ~ 1 ) x ~ + ~ - ~ ) 2

o ~ - x j - p o ~ -

x j + l )

_ _ U - d - U + x ( X j - - O l ) -'~ l U x x ( ~ 2 ) ( X j - -

O~)2 __ Ztx ,

o ~ - x a - p c ~ -

X j + l )

w here ~ 1 E

(o~ ,x j+l )

a n d ~2 E ( x j , o ~ ) . W i t h t h e j u m p c o n d i t i o n s u + = u - a n d

u + - - pu x ,

t h e

e x p r e s s i o n a b o v e i s s i m p l i f i e d t o





260

Z . L i / A p p l i e d

Numer i ca l M a thema t i c s

27 (1998) 253-267

P r o o f . I f c~ ~ [ X i , X i + l , t h e n / 3 x ) a n d

v l h x ) a r e

t w o c o n s t a n t s . S o

Xi+l Xi+l

x i x i

g,t

z z _~

/ ~ ( X i + I / 2) V h ( X i + I / 2 ) ( U ( X ) __ U I ( X ) )

x i + ,

x i

w h e r e X i + l / 2 = X i

J r h / 2 .

O n t h e o t h e r h a n d i f o L [ X j , X j + I , t h e n

Xj4-1

f ~ ( X ) ( U ( X ) - t t i ( x ) ) t V l h ( X ) d x

x j

oL Xj+ l

~---~ - V S ,( ~ 'l) / U ( X ) -

Z I ( X ) )

d x

- { - / ~ - t - v ~ ( ~ C 2 )

/ ( ' U ( X ) -

g l ( X ) ) t

d x

x j

o~

= 9 - v g ( ¢ , ) ( ~ - - * 6 ) - ; ~ ~ ( ¢ 2 ) ( ~ + - ~ / + ) = 0 ,

w h e r e { 1 a n d { 2 a r e a n y t w o p o i n t s i n t h e i n t e r v a l

x j , c~ )

a n d ~ ,

x j + l ) ,

r e s p e c t i v e l y . I n t h e d e r i v a t i o n

a b o v e , w e h a v e u s e d t h e n a tu r a l j u m p c o n d i t i o n 1 . 3 ) , 1 . 4 ) f or t h e b a s i s f u n c t i o n

V h

a n d c o n t i n u i t y

c o n d i t i o n s f o r u x ) a n d u z x ) at c~:

- -

/~ V h ( ~ l ) = / ~ + / ) h ( ¢ 2 ) , t t + = i t - , U t = U 7 . [ ]

B e l o w i s t h e m a i n t h e o r e m o f c o n v e r g e n c e f o r t h e m o d i f i e d f i n i t e e l e m e n t m e t h o d .

T h e o r e m 3 . 4 .

L e t U h X ) b e th e so l u t i o n o b t a i n e d f r o m t h e f i n i t e e l e m e n t m e t h o d w i t h t h e m o d i f i e d

b a s i s f u n c t i o n . T h en

I I U ( X ) - U h ( X ) llo ~

Cllu"llooh 2, (3.31)

w h e r e

C - 2 m a x { l , p } 3

+ 2 3 . 3 2 )

m i n { 1 , p }

P r o o f . F o r a n y v h C V h , w e h a v e

a(u

- t t i , V h ) =

a(u

- t t h q- t t h - u I~ V h) :

a(u -

Uh, Vh) Jr-

a(uh

- - t t I ~ Vh ) "=

a(uh

- - U I ~ V h ) .

F r o m t h e d e f i n i t i o n o f a u , v ) a n d L e m m a 3 . 3, w e k n o w t ha t

1 N - 1 X i + l

0 i = 0 x i

T a k e

V h = U h - - U 1 E V h ,

w e c o n c l u d e

a u h - - u i , U h

- - U X ) = 0 , w h i c h i m p l i e s t h a t

U h X ) = u r x ) .

T h u s

t , ~ ( ~ ) - ~ h ( X ) [ < - 1 ' 4 ) - - ~ * ' ( x ) l + 1 ~ I ( x ) - - ~ h ( X ) l - - < I ' * ( x ) - - u ~ ' ( x ) l ~ < C P I ~ I I ~

h 2 .

[ ]



Z. Li /Applied NumericalMathematics27 (1998) 253-267

261

R e m a r k 3 .1 . T h e c o n c l u s i o n c a n b e e a s i ly e x t e n d e d to t h e m o r e g e n e r a l c a s e (1 .1 ) , (1 .2 ) w i t h v a r ia b l e

/ 3 (x ) . T he key m od i f i c a t i on i s t he fo l l ow i ng :

N - I x i + l

1 .

a ( u - u i , Vh)= ~ / ( /3 (x) - ~+ ~/2) (u

U i ) t V t h

d

i = O , i ¢ j , j + l x i

dx

O~ Xi+l

+ f (/ 3 (x )- -~l)(u - uz)'Vlh dx + / (/ 3 (x )- -~r)(U - UI)'Vlh dx,

x j a

w h e r e

~i+1/2, ~l

and ~ r a r e the ave ra ge va l ue s o f / 3 (x ) i n t he i n te rva l s [x i , x i + l ] ,

[xj,

c~] and [c~,

xj+l],

r e spec t i ve l y , w h i ch a r e f i r st o rde r app rox i m a t i ons t o / 3 ( x ) i n t ha t spec if i c i n t e rva l . N o t e t ha t

ui (x )

is

a s e c o n d o r d e r a p p r o x i m a t i o n t o u(x) i n t he i n f i n i t e no rm , w h i ch m eans t ha t u)(x) i s a f i r s t o rder

app rox i m a t i on t o

u~(x)

excep t a t g r i d po i n t s and t he i n t e r f ace c~ . T hus w e have

a(uh -- ui, Vh) <. Clh llu -

u / I l l

I l v h l l l ~ < Clh211u "ll~llvhlll.

T a k i n g Vh = Uh -- UI E Vh, w e c o n c l u d e

II ~h - ~ 1 1 1 < c l h 2 1 1 ~ l [ ~ .

T he f i na l i nequa l i t y t hen fo l l ow s :

l u ( x ) - U h ( X ) l ~< l u ( x ) - u ~ ( x )l + l u i ( x ) - u h ( z ) l

< +

~ C I l ~ " l l~ h 2 ,

w i t h a d i f f e r en t e r ro r cons t an t C .

4 Num er ica l exam ples

We have done qu i t e a num ber o f num er i ca l t e s t s . A l l t he r e su l t s ag ree w i t h t he t heo re t i c a l ana l y s i s .

T he i n t eg ra l s

x i + l x i + l

/ ¢ i ( x ) f ( x ) d x

and /

¢:(x)¢1j(x)dx

x i x i

a re eva l ua t ed u s i ng t he t r apezo i da l ru l e . We j u s t p r e sen t one exam pl e be l ow .

T he d i f f e r en t i a l equa t i on i s

/3 - if x < c~,

(/3Ux)x=

1 2 x 2, 0 ~ < x ~ < 1 , / 3 = /3 + i f x > ~ ,

~ ( 0 ) = o , ~ ( 1 ) = 1 / 9 + + ( 1 / 9 - - 1 / 9 + ) ~ 4 .

T he na t u ra l j um p cond i t i ons (1 . 3) , ( 1 . 4 ) a r e s a t is f ied . T he e xac t so l u t i on is



262

Z. Li /App lied Numerical Mathematics 27 (1998) 253-267

f X 4 / / 3 - i f x < c~,

u(x) = I x4//3+ +

(1 //3- - 1/ /3+ )c0 if x > c~,

where the parameters/3- and/3+ are two constants.

Since Simpson 's rule has degree of precision three and f ( x ) is a quadratic, there are no errors

in computing

f ¢ i ( x ) f ( x )dx

and f ¢~(x)¢}(x)dx. We expect the solution obtained using the FEM

method to be the same as the interpolating function defined in (3.21), provide there is no round-off

error involved. In other words, the compu ted solution agrees with the exact solution at grid points and

is second order accurate at any other points. Numerical experiments have confirmed the theoretical

analysis. The infinity norm o f the computed solution at grid points is between 6 x 10 -15 to 3 x 10 -13

in double precision. At any other points, the FEM solution is defined as

N

Uh(X) = ~ ui¢i(x), (4.33)

i=0

where ui is the computed solution at the grid point xi, the error decreases by a factor of 4 if we

double the grid size. Table 1 shows the grid refinement analysis in the infinity norm for two different

2

points which are not part of the grid. In the first ca se ,/ 3- = 1,/3+ = 100, and the interface is o~ =

which is not a grid point. We see that the computed solution at the interface itself has average second

order accuracy. In the second case, we take the sam e/ 3- and /3+ , but the interface is c~ = 0.5 which

is a grid point. Since there is no interface between the grid points, the solution at the interface c~ is

again accurate to the machine precision up to a factor of the condition number of the discrete linear

system. The r ight part of the table shows the grid refinement analysis at c~ + ~ which is not a grid

point. We see that the error is reduced by a factor of 4. Notice that, for interface problems, the error

constant which is O(1) may not approach to a constant. It will depend on the relative position of the

interface and the grid. This is the case in the left part of the table. By the second order accuracy, we

actua lly mean the average convergence rate of the solution, the reader is referred to [10,12] for more

information on the error analysis. For the second case, since the interface is a grid point, the error

constant will indeed approach to a fixed number.

Fig. 3(a) is the plot of the solution with 40 grid points. There is no difference between the computed

and the exact solution at the grid points. The differences in other places are too small to be visible.

Table 1

Grid refinement analysis for the examp le with /3- = 1, /3+ = 100. The left part of the table: the error of the solution

2 The right part of the table: the rr or of the soluti on evalu ated at x = 0.5 + 5' c~ = 0.5valua ted at the inter face x = c~ = 7

7~, Cn en /e 2n Cn/ C4 n '1% Cn en /C 2n

20 4.4312 × 10 -5

40 5.4822 × 10 -6 8.0829

80 2.7347 × 10 -6 2.0047

160 3.4478 × 10 7 7.9318

320 1.7038 × 10 -7 2.0235

640 2.1582 × l0 -8 7.8948

16.2038

15.9010

16.0503

15.9752

20 2.2844 × 10 -5

40 5.8259 × 10 -6 3.9211

80 1.4420 × 10 -6 4.0403

160 3.6229 × 10 -7 3.9801

320 9.0347 x 10 -8 4.010 0

640 2.2615 × 10 -8 3.9950



Z. Li /Ap plied Numerical Mathematics 27 (1998) 253-267 263

0 3

0 2 5

0 2

0 1 5

0 1

0 0 5

0

0 0 5

n = 4 0 , [~ = 1 [ ~ + = 1 0 0 f~2/3

0 0 C o m p u t e d

E x a c t

C = = C C C O ~

I I I I I I t I I

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9

× 1 4

E r r o r P l o t n : 4 0 , ~ 1 , { ~ * : 1 0 0 1 1 -- 2/ 3

(a) (b)

Fig. 3. Com parison of the com puted solution and exact solution when ~z = 40. (a) The solution plo t. (b) The erro r plot.

F i g . 3 ( b ) i s t h e e r r o r p l o t o f t h e e r r o r i n th e e n t i r e i n t e r v a l . W e s e e t h e e r r o r s a r e z e r o a t g r i d p o i n t s

and

O ( h 2 ) a t o t h e r p o i n t s .

5 C o n s t r u c t i n g b a s i s f u n c t i o n s i n t w o d i m e n s i o n s

T h e m o d e l e q u a t i o n i n t w o d i m e n s i o n s o n a r e c t a n g u l a r r e g i o n w i t h a c l o s e d i n t e r f a c e i s

V . ( / 3 V u ) = f ( x , y ) , ( x , y ) c f2 , ( 5 . 3 4 )

g i v e n B C o n 0 S ?, ( 5 .3 5 )

s e e th e d i a g r a m i n F i g . 4 . T h e n a t u r a l j u m p c o n d i t i o n s a c r o s s th e i n t e r f a c e F a r e

[u] = 0, [/3u~] = 0, (5 .36 )

w h e r e u ~ i s t h e n o r m a l d e r i v a t i v e .

A s d i s c u s s e d i n p r e v i o u s s e c t i o n s , w e w i l l u s e a u n i f o r m t r i a n g u l a t i o n , s e e F ig . 5 . I f a c e l l c o n t a i n s

n o i n t e r f a c e, w e c a n u s e t h e s t a n d a r d l i n e a r b a s is f u n c t i o n o v e r t h a t c e l l. I t i s m o r e d i f f ic u l t t o c o n s t r u c t

b a s i c f u n c t i o n i n t w o d i m e n s i o n s w h e n i n t e r f a c e c u t s t h r o u g h t h e u n i f o r m t r i a n g u l a t i o n . I t i s t r u e t h a t

w e c a n e a s i l y f i n d p i e c e w i s e l i n e a r , o r q u a d r a t i c , o r c u b i c f u n c t i o n w h i c h i n t e r p o l a t e s t h e s o l u t i o n

o f ( 5 . 3 4 ), ( 5 .3 5 ) t o s e c o n d o r h i g h e r o r d e r a c c u r a c y u s i n g t h e T a y l o r e x p a n s i o n . T h e d i f f i c u l t y i n

c o n s t r u c t i n g t h e b a s i s f u n c t i o n i s t h e r e q u i r e m e n t s o f c o n t i n u i t y in t h e e n t ir e r e g i o n a n d t h e j u m p

c o n d i t i o n s a c r o s s t h e i n t e r f a ce . T h e r e a r e s e v e r a l a p p r o a c h e s c u r r e n t l y u n d e r i n v e s t i g a t io n . B u b e a n d

K a u p e [ 2 ] a r e t y i n g t o u s e a q u a d r i l a t e r a l t r i a n g u l a t i o n . H o u a n d W u [ 5 ] a r e e x p e r i m e n t i n g w i t h b o t h

c o n f o r m i n g a n d n o n - c o n f o r m i n g b a s i s f u n c t i o n s . B e l o w w e p r o p o s e a n a p p r o a c h w h i c h i s i n t h e s a m e

s p ir it a s o u r d i s c u s s i o n f o r o n e d i m e n s i o n a l p r o b l e m . W e w i l l s ti c k w i th c o n f o r m i n g b a s is f u n c t io n s ,

t h a t i s , t h e b a s i s f u n c t i o n s b e l o n g t o H01 ( £ 2) .



264 Z Li /Appli ed Numerical Mathematics 27 (1998) 253-267

~+

f~

F i g . 4. A d i a g r a m f o r a m o d e l p r o b l e m i n t w o d i m e n s i o n s .

F

A~

E

y~

G

h

xl

Fig . 5 . A typ ica l c e l l w i th the in t e r face cu t t ing th rough .

5.1. A coupled approach

N o w t a k e a t y p i c a l c a s e a s s h o w n i n F i g . 5 . A n a r b i t r a r y c l o s e d i n t e r f a c e c a n b e a p p r o x i m a t e d b y

p i e c e w i s e li n e s e g m e n t s . W e w a n t t o f in d a b a s i s f u n c ti o n ¢ ( x , y ) , f o r e x a m p l e , c e n t e r e d a t

(xi , y j)

w h i ch s a t i s f i e s

¢ ( x , y ) E C ( O ) ,

¢(x i , y j )

= l , (5 .37)

¢ ]

= o , = o . 5 . 3 8 )

I t i s q u i t e o b v i o u s th a t a li n ea r b a s i s f u n c t i o n w i l l n o t w o r k . I n s t ead , w e t r y to u s e a p i ecew i s e q u ad r a t i c

b as i s f u n c t i o n . I n F i g . 5 , t h e i n t e r f ace d o es n o t cu t t h r o u g h t h e t r i an g l e s 4 , 5 an d 6 . T h e r e f o r e t h e

b as i s f u n c t i o n can b e t ak en a s t h e s t an d a r d l i n ea r b a s i s f u n c t i o n o v e r t h o s e t r i an g l e s .

T h e t r i an g l e s l , 2 an d 3 co n t a i n a p o r t i o n o f t h e i n t e rf ace w h i ch d i v i d e s t h e r eg i o n i n t o s i x p i ece s ,

t h r ee tr i an g l e s an d t h r ee q u ad r i la t e r a l s. T h e p i e cew i s e q u ad r a t i c f u n c t i o n o v e r th e s i x p i ece s can b e

d e t e r m i n e d u s i n g t h e

undetermined coefficient method.

T h e t o t a l n u m b e r o f d e g r e e s o f f r e e d o m i s 3 6

w i t h o u t an y co n s t ra i n s . T h e co n t i n u i t y co n s t r a i n s i n v o l v i n g b o t h t h e s i d e s o f t h e tr i an g l e s an d t h e

i n te r f a ce a r e 3 3 . T h e r e m a i n i n g 3 d e g r e e s o f f r e e d o m a r e t h en u s e d t o s a t is f y th e f l u x j u m p c o n d i t i o n

[ /3¢n] --- 0 to cer t a in de gree . Fo r ins tanc e , we can f orce [ /3¢n] to be zero a t the m id-po in t s o f l inear



Z L i / A p p l i e d N u m e r ic a l M a t h e m a t i c s 2 7 1 9 9 8 ) 2 5 3 - 2 6 7

265

s e g m e n t s

A B , B C ,

a n d

C D .

A t o t h e r p o i n t s o f t h e i n t e r face , I t C h ] w i l l a l s o b e v e r y s m a l l f r o m t h e

c o n t i n u i ty c o n d i t i o n o f t h e s o l u t i o n a n d t h e a s s u m p t i o n t h a t t h e n o r m a l d i r e c ti o n o f t h e i n t e rf a c e d o e s

n o t c h a n g e t o o m u c h i n t h e c e l l .

I n t h e en d , w e n eed t o s e t u p a l in ea r s y s t em o f eq u a t i o n f o r t h e co e f f i c i en t s o f t h e q u ad r a t i c b a s i s

f u n ct io n . T h e n u m b e r o f u n k n o w n s c a n b e g r e at ly r e d u c e d i f w e t a k e a d v a n t a g e o f th e b o u n d a r y

co n d i t i o n s . F o r ex am p l e , t h e f u n c t i o n ¢ ( x , y ) o n th e q u ad r i l a t e ra l

A B O E

can b e w r i t t en a s

¢ ( x , y ) = a o + a l ( x - x i - 1 ) + a 2 (y - - y j ) + a 3 ( x - - x i - 1 ) ( y - y j ) .

I t i s n o t s o e a s y t o i m p l e m e n t t h e a p p r o a c h d i s c u s s e d a b o v e b e c a u s e t h e b a s i s f u n c t i o n s o n s e v e r a l

ce l l s a r e co u p l ed t o g e t h e r . T h i s i s t h e p r i ce w h i ch w e n eed t o p ay f o r i n t e r f ace p r o b l em s t o o b t a i n

m o r e a c c u r a t e r e s u lt s . H o w e v e r w e c a n s i m p l i f y t h e p r o c e s s o f c o n s tr u c t in g t h e b a s is f u n c t i o n i f w e

c a n

p r e - d e t e r m i n e

t h e v a l u e s o f t h e b a s i s f u n c t i o n a t p o i n t s B , an d C a s in th e ex a m p l e o f F i g . 5 . A n

i n t e r p o l a t i o n ap p r o ach t o d e t e r m i n e t h o s e v a l u e s w i l l b e d i s cu s s ed l a t e r i n t h i s s ec t i o n .

5 .2 . A d eco u p l ed a p p ro a ch

S u p p o s e w e c a n p r e - d e t e r m i n e t h e v a l u e s o f th e b a s i s f u n c ti o n a t th o s e p o i n t s B , a n d C i n F i g . 5 ,

t h en w e can co n s t r u c t a b i l i n ea r f u n c t i o n i n each q u ad r i l a t e r a l w h i ch i n t e r p o l a t e s t h e f u n c t i o n v a l u e s

a t t h e v e r t i c e s an d t h o s e i n t e r s ec t i o n s s u ch a s t h e p o i n t s B , an d C i n F i g . 5 . T h e b i l i n ea r f u n c t i o n s

¢ ( x , y ) w i l l b e l i n e a r o n t h e b o u n d a r y o f th e r e g i o n a s s h o w n i n F ig . 5 . F o r e x a m p l e , t h e b i l in e a r

f u n c t i o n ¢ ( x , y ) o n t h e q u ad r i l a t e r a l A B O E is

¢ X , y ) - - X - h x i - 1 q - ~ Y - y j ÷ 1 ÷ q ( x - x i - 1 ) y - y f i ) ,

w h e r e q i s c h o s e n s u c h t h at ¢ ( x B , YB ) = C B , th e p r e - d e t e r m i n e d v a l u e o f t h e b a si s f u n c t i o n a t B .

O n c e w e h a v e d e t e r m i n e d t h e b i l i n e a r f u n c t i o n o n e a c h q u a d r i l a t e r a l , t h e n w e k n o w t h e v a l u e s

o f th e b a s i s f u n c t i o n a t m i d - p o i n t s o f a ll s i d e s o f e ach t r i an g l e s. T h u s a q u ad r a t i c f u n c t i o n i s e a s i l y

d e t e r m i n ed f r o m t h e s i x v a l u e s o f th e b a s i s f u n c t i o n o n each t r i an g l e .

I n t h e a p p r o a c h w e d e s c r i b e d a b o v e , w e a r e a l m o s t a b l e t o f i n d t h e b a s i s f u n c t i o n o n e a c h t r i a n g l e

s ep a r a t e l y , w h i ch m ak es i t e a s i e r t o a s s em b l e t h e s t i f f n e s s m a t r i x .

5 .3 . I n t e rp o l a t io n sch em e f o r t h e p re -d e t e rm i n ed va l u e s

T h e ap p r o ach d e s c r i b ed ab o v e r e l y o n t h e v a l u e s a t i n t e r s ec t io n s o f t h e i n t e r face an d t h e i n te r i o r

s i d e s o f t h e t ri an g l e s . I d ea l l y , t h e b a s i s f u n c t i o n can b e t ak en a s t h e s o l u t i o n o f th e f o l l o w i n g P o i s s o n

e q u a t i o n w i t h n a t u r a l j u m p c o n d i t i o n s :

v . ( 3 v ¢ ) = o ,

[¢] = o , [9 ¢ , ] = o .

T h e b o u n d a r y c o n d i t i o n i s

I X - - X i - - 1

X - - X i

¢ =

0

o n

E O ,

o n

E F ,

o n E G , G H a n d H F .

( 5 . 3 9 )

( 5 . 4 0 )

(5 .41)



266 Z.

Li / Applied Numerical Mathematics 27 (1998) 253-267

O n c e w e k n o w t h e s o l u t io n o f th e P D E a b o v e , w e c a n g e t t h o s e p r e - d e t e rm i n e d v a l u e s . H o w e v e r , it

i s t o o co s t l y t o s o l v e t h e P D E ab o v e a t a l l t h e ce l l s w h i ch co n t a i n t h e i n t e r f ace . W h a t w e can d o ,

h o w e v e r , is t o u s e a n i n te r p o l a ti o n s c h e m e i n t e rm s o f t h e b o u n d a r y v a l u e s , t h e j u m p c o n d i t io n s , a n d

t h e p a r t i a l d i f f e r en t i a l eq u a t i o n i t s e l f , f o r ex am p l e ,

T h e co e f f i c i en t s c an b e d e t e r m i n ed u s i n g t h e w e i g h t ed l e a s t s q u a r e s i n t e r p o l a t i o n [ 1 0] an d t h e p r o ced u r e

i s b r i e f l y d e s c r i b ed b e l o w :

S e l ec t a p o i n t X o n t h e i n t e r face .

. . . . .+

U s e t h e l o ca l co o r d i n a t e s a t X i n t h e t an g en t i a l an d n o r m a l d i r ec t io n s o f t h e i n t e r f ace .

_____+*

U s e t h e T a y l o r e x p a n s i o n o v e r X t o e x p a n d t h e v a l u e s f r o m e a c h s i d e o f t h e i n te r fa c e .

E l i m i n a t e t h e q u an t i ti e s o f o n e s i d e , s u ch a s th e s o l u t i o n , th e d e r i v a t i v e s u p t o s eco n d o r d e r, i n

t e r m s o f an o t h e r u s i n g t h e j u m p c o n d i t i o n s a n d t h e d i f f e ren t i a l eq u a t i o n .

S e t u p an d s o l v e t h e li n ea r s y s t e m o f eq u a t i o n t o g e t t h e co e f f i c i en t s o f th e i n t e r p o l a ti o n .

T h e d e t a i l s c an b e f o u n d i n [ 8 , 1 0 ] w i t h s o m e m o d i f i c a t i o n .

T h eo r e t i c a l l y , i f t h e b a s i s f u n c t i o n s b e l o n g t o H 01 ( ~ ) s p ace an d s a t i s f y t h e n a t u r a l j u m p c o n d i t i o n s ,

t h e n t h e s t a n d a r d e r r o r a n a l y s i s u s i n g t h e e n e r g y n o r m w o u l d a p p l y . S o w e w o u l d h a v e t h e s t a n d a r d

co n v e r g en c e r e s u l t ev e n i f t h e r e i s an i n t e r f ace i n th e s o l u t i o n d o m a i n . I t is n o t e a s y t o s ee o r p r o v e

w h e t h e r t h e m e t h o d s d e s c r i b e d a b o v e a r e s t i l l s e c o n d o r d e r a c c u r a t e i n t h e i n f i n i t y n o r m . H o w e v e r ,

t h e a p p r o a c h e s d e s c r i b e d a b o v e c e r t a i n l y h a s b e t t e r a c c u r a c y c o m p a r e d t o t h e s t r a i g h t f o r w a r d f i n i t e

e l em en t m e t h o d w i t h n o m o d i f i c a t i o n s . T h e p r i ce i s t h e ex t r a co s t a t t h o s e ce l l s w h e r e t h e i n t e r f ace

cu t s t h r o u g h .

I n s u m m ar y , t h e m o d i f i ed f in i te e l em en t m e t h o d u s i n g t h e s i m p l e o r u n i f o r m t r i an g u l a t i o n i s v e r y

a c c u r a t e f o r o n e - d i m e n s i o n a l p r o b l e m s a n d v e r y p r o m i s i n g f o r t w o - d i m e n s i o n a l p r o b l e m s . T h e c o r -

r e s p o n d i n g f in i te d i ff e r e n c e m e t h o d w o u l d a l l o w u s to d e a l w i th i n h o m o g e n e o u s j u m p c o n d i t io n s .

H o w e v e r , t h e a n a l y s i s f o r t w o o r h i g h e r d i m e n s i o n a l i n t e r f a c e p r o b l e m s i s f a r f r o m c o m p l e t e . W e

h o p e t h e i d ea s p r e s en t ed i n t h i s p ap e r w i l l ev en t u a l l y l e ad t o t h e d ev e l o p m en t o f s o m e e f f i c i en t f i n i t e

e l e m e n t m e t h o d s w i t h s i m p l e t ri a n g u la t io n s f o r t w o a n d t h r ee d i m e n s i o n a l p r o b l e m s .

cknowledgements

T h e a u t h o r w o u l d li k e to th a n k T h o m a s H o u , T o n y C h a n , J i n c a o X u , X i a o h u i W u a n d o t h e r p e o p l e

f o r v e r y u s e f u l d i sc u s s i o n s a n d e n c o u r a g e m e n t s f o r t h is p r o j e c t s. T h e a u t h o r a l s o t h a n k s t h e r e f e r e e ' s

c o m m e n t s a n d s u g g e s t i o n s .

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